Moving Block Bootstrap Research Articles

Convergence rates of empirical block length selectors for block bootstrap

We investigate the accuracy of two general non-parametric methods for estimating optimal block lengths for block bootstraps with time series – the first proposed in the seminal paper of Hall, Horowitz and Jing (Biometrika 82 (1995) 561–574) and the second from Lahiri et al. (Stat. Methodol. 4 (2007) 292–321). The relative performances of these general methods have been unknown and, to provide a comparison, we focus on rates of convergence for these block length selectors for the moving block bootstrap (MBB) with variance estimation problems under the smooth function model. It is shown that, with suitable choice of tuning parameters, the optimal convergence rate of the first method is $O_{p}(n^{-1/6})$ where $n$ denotes the sample size. The optimal convergence rate of the second method, with the same number of tuning parameters, is shown to be $O_{p}(n^{-2/7})$, suggesting that the second method may generally have better large-sample properties for block selection in block bootstrap applications beyond variance estimation. We also compare the two general methods with other plug-in methods specifically designed for block selection in variance estimation, where the best possible convergence rate is shown to be $O_{p}(n^{-1/3})$ and achieved by a method from Politis and White (Econometric Rev. 23 (2004) 53–70).

Estimating the time-varying NAIRU and the Phillips curve slope simultaneously: a note

This article presents a new method for estimating the unobservable nonaccelerating inflation rate of unemployment (NAIRU). We improve upon the method employed in Ball and Mankiw (2002) so that (i) the new method can estimate simultaneously both the time-varying NAIRU and the Phillips curve slope and (ii) it can yield traditional constant NAIRU estimates as an extreme. As an empirical illustration, we estimate the time-varying US NAIRU with the moving block bootstrap confidence intervals.

New second order cyclostationary analysis and application to the detection and characterization of a runner׳s fatigue

Cyclostationarity (CS), as characteristic of signals, is a technique that offers diagnostic advantages for the analysis of failures, faults and disturbances which are related to a system being examined. The aim of this paper is to introduce the concept of CS for signals and to present possibilities of statistical resampling procedures for the estimation of second order statistics of such signals. The resampling methods treated in this paper are referred to as subsampling and moving block bootstrap (MBB). A description of these methods is presented and their applicability to CS simulated data is proved. The comparison between those two procedures shows that subsampling seems to be more efficient and more relevant than MBB. The subsampling-based CS analysis of biomechanical ground reaction force signals (GRF signals), coming from a high level professional runner, proves the second order cyclostationary (CS2) nature of the GRF signals. Moreover, it enables us to distinguish successfully the CS2 cyclic frequencies of the signals under consideration. Some new indicators based on the subsampling procedure are also proposed. This allows a better characterization and a full innovative description of the different fatigue states of a runner.

More accurate, calibrated bootstrap confidence intervals for estimating the correlation between two time series

Estimation of Pearson’s correlation coefficient between two time series, in the evaluation of the influences of one time-dependent variable on another, is an often used statistical method in climate sciences. Data properties common to climate time series, namely non-normal distributional shape, serial correlation, and small data sizes, call for advanced, robust methods to estimate accurate confidence intervals to support the correlation point estimate. Bootstrap confidence intervals are estimated in the Fortran 90 program PearsonT (Mudelsee, Math Geol 35(6):651–665, 2003), where the main intention is to obtain accurate confidence intervals for correlation coefficients between two time series by taking the serial dependence of the data-generating process into account. However, Monte Carlo experiments show that the coverage accuracy of the confidence intervals for smaller data sizes can be substantially improved. In the present paper, the existing program is adapted into a new version, called PearsonT3, by calibrating the confidence interval to increase the coverage accuracy. Calibration is a bootstrap resampling technique that performs a second bootstrap loop (it resamples from the bootstrap resamples). It offers, like the non-calibrated bootstrap confidence intervals, robustness against the data distribution. Pairwise moving block bootstrap resampling is used to preserve the serial dependence of both time series. The calibration is applied to standard error-based bootstrap Student’s $$t$$ confidence intervals. The performance of the calibrated confidence interval is examined with Monte Carlo simulations and compared with the performance of confidence intervals without calibration. The coverage accuracy is evidently better for the calibrated confidence intervals where the coverage error is acceptably small already (i.e., within a few percentage points) for data sizes as small as 20.

On estimation of almost ideal demand system using moving blocks bootstrap and pairs bootstrap methods

This paper applies a bootstrap method to the Almost Ideal Demand System (AIDS) proposed by Deaton and Muellbauer (Am Econ Rev 70:312–326, 1980), where the moving blocks bootstrap (MBB) and pairs bootstrap (PB) methods are adopted taking into account serially correlated error terms and limited dependent variables (note that the dependent variables in the AIDS model lie on the interval between zero and one). We aim to obtain the empirical distribution of the expenditure and price elasticities. Note that, the expenditure and price elasticities are obtained using the parameter estimates included in the AIDS model. In the past, a few studies report both the elasticity estimates and their standard errors obtained from the Delta method, but most of studies show only the elasticity estimates (i.e., statistical tests have not been done in most of the past studies). Applying MBB and PB methods to the AIDS model and using Japanese monthly household expenditure data from January, 1975 to December, 2012, we show in this paper that a few elasticities are statistically insignificant. We also compare the standard errors based on the bootstrap method with those based on the Delta method. We obtain the results that the differences between the Delta method and the bootstrap method are not negligible. In addition, the validity of the linear approximated AIDS (LA–AIDS) model which is commonly used in empirical studies is examined. In consequence, we find that the LA–AIDS model shows a poor performance, compared with the AIDS model, because the LA–AIDS model yields inconsistency on the elasticity estimates.

Resampling Methods for Time Series Level Crossings

We present an application of subsampling and bootstrap methods for time series to determine the distribution of the estimator of zero crossings. The zero crossings method provides an alternative estimator of the lag-1 autocorrelation coefficient that is reducing the data storage requirements and is more robust with respect to outliers when compared to the classical estimator. The main results here are showing the consistency of subsampling, the consistency of moving block bootstrap, the consistency of non overlapping block bootstrap and the consistency of stationary bootstrap for this estimator. Theorems are formulated for Gaussian processes, elliptically symmetric processes and processes which are transformed Gaussian processes. Theoretical results are illustrated by simulations and practical data analysis. We have also shown that in practice the MBB method behaves better than the subsampling method.

Ensemble statistical post-processing of the National Air Quality Forecast Capability: Enhancing ozone forecasts in Baltimore, Maryland

An ensemble statistical post-processor (ESP) is developed for the National Air Quality Forecast Capability (NAQFC) to address the unique challenges of forecasting surface ozone in Baltimore, MD. Air quality and meteorological data were collected from the eight monitors that constitute the Baltimore forecast region. These data were used to build the ESP using a moving-block bootstrap, regression tree models, and extreme-value theory. The ESP was evaluated using a 10-fold cross-validation to avoid evaluation with the same data used in the development process. Results indicate that the ESP is conditionally biased, likely due to slight overfitting while training the regression tree models. When viewed from the perspective of a decision-maker, the ESP provides a wealth of additional information previously not available through the NAQFC alone. The user is provided the freedom to tailor the forecast to the decision at hand by using decision-specific probability thresholds that define a forecast for an ozone exceedance. Taking advantage of the ESP, the user not only receives an increase in value over the NAQFC, but also receives value for costly decisions that the NAQFC couldn't provide alone.

On the performance of block-bootstrap continuously updated GMM for a class of non-linear conditional moment models

In the context of the continuously updated generalized-methods-of-moments (GMM), this study evaluates the finite sample properties of Wald- and criterion-based bootstrap inference for a class of models defined by non-linear conditional moment functions. This work provides simulation evidence that validates the moving block-bootstrap (MBB) as an alternative to asymptotic approximation for robust finite sample GMM inference. The study considers data generating processes with highly non-linear conditional moment functions, weak instruments, and near failure of the identification condition. In the absence of a consensus on best practice when identification is weak, Monte Carlo results of this study are encouraging to the empirical researchers. For criterion-based tests, the MBB performs fairly well in reducing the error in the rejection frequency that occurs when first-order asymptotic critical values are used. In particular, it is possible to improve finite sample inference by inverting bootstrap Wald-type statistics which are commonly used in practice The bootstrap percentile- $$t$$ t confidence intervals performed better than the asymptotic confidence intervals but only marginally in weakly identified specifications with high non-linear moment functions.

Testing cointegration relationship in a semiparametric varying coefficient model

In this paper, we develop two cointegration tests for two varying coefficient cointegration regression models, respectively. Our test statistics are residual based. We derive the asymptotic distributions of test statistics under the null hypothesis of cointegration and show that they are consistent against the alternative hypotheses. We also propose a wild bootstrap procedure companioned with the continuous moving block bootstrap method proposed in Paparoditis and Politis (2001) and Phillips (2010) to rectify severe distortions found in simulations when the sample size is small. We apply the proposed test statistic to examine the purchasing power parity (PPP) hypothesis between the US and Canada. In contrast to the existing results from linear cointegration tests, our varying coefficient cointegration test does not reject that PPP holds between the US and Canada.

Maximum Entropy Bootstrap Simulations for Variance Estimation

We report a simulation study where we want to consider a fairly simple data generating process (DGP) used in Nordman and Lahiri (JASA, 2012) with a single fixed regressor and regression errors produced by simple AR(1) processes. We focus on the estimation of standard errors of regression coefficients, not the coefficients themselves. We compare confidence intervals by three inference procedures: the usual Chi-square distribution (Chi-sq), the moving blocks bootstrap (MBB) and a newer maximum entropy bootstrap (meboot). Since simulations have a known true standard error, we can assess the coverage and consistency of the meboot. The traditional Chi-sq confidence intervals have very poor coverage, suggesting that they should not be used in the presence of auto-correlated errors. We also consider the advisability of symmetrizing transformation of the ME density by repeating the experiments. We find that symmetrizing offers a slight advantage. Since the meboot appears to be generally superior to others, it can be recommended.

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In the simulation output analysis, bootstrap method is an applicable resampling technique to insufficient data which are not significant statistically. The moving block bootstrap, the stationary bootstrap, and the threshold bootstrap are typical bootstrap methods to be used for autocorrelated time series data. They are nonparametric methods for stationary time series data, which correctly describe the original data. In the simulation output analysis, however, we may not use them because of the non-stationarity in the data set caused by the trend such as increasing or decreasing. In these cases, we can get rid of the trend by differencing the data, which guarantees the stationarity. We can get the bootstrapped data from the differenced stationary data. Taking a reverse transform to the bootstrapped data, finally, we get the pseudo-samples for the original data. In this paper, we introduce the applicability of bootstrap methods to the time series data having trend, and then verify it through the statistical analyses.

Maximum Entropy Bootstrap Algorithm Enhancements

While moving block bootstrap (MBB) has been used for mildly dependent (m-dependent) time series, maximum entropy (ME) bootstrap (meboot) is perhaps the only tool for inference involving perfectly dependent, nonstationary time series, possibly subject to jumps, regime changes and gaps. This brief note describes the logic and provides the R code for two potential enhancements to the meboot algorithm in \citet{VinodJavier:2009}, available as the meboo package of the R software. The first rescaling adjusts the of meboot resampled elements so that the population variance of the ME density equals that of the original data. Our second forces the ME density to be symmetric. One simulation involving inference for regression standard errors suggests that the symmetrizing enhancement of the meboot continues to outperform the MBB.

Moving Block Bootstrap for Analyzing Longitudinal Data

In a longitudinal study subjects are followed over time. I focus on a case where the number of replications over time is large relative to the number of subjects in the study. I investigate the use of moving block bootstrap methods for analyzing such data. Asymptotic properties of the bootstrap methods in this setting are derived. The effectiveness of these resampling methods is also demonstrated through a simulation study.

Valid Resampling of Higher-Order Statistics Using the Linear Process Bootstrap and Autoregressive Sieve Bootstrap

We show that the linear process bootstrap (LPB) and the autoregressive sieve bootstrap (AR sieve) are, in general, not valid for statistics whose large-sample distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or not. Inspired by the block-of-blocks bootstrap, we circumvent this non-validity by applying the LPB and AR sieve to suitably blocked data and not to the original data itself. In a simulation study, we compare the LPB, AR sieve, and moving block bootstrap applied directly and to blocked data.

Unstructured Road Spectrum α-stable Distribution Parameters Interval Estimation and Reconstruction Based on Moving Block Bootstrap Method

Symmetrical α-stable distribution is used to express the amplitude non-Gaussian characteristics of the unstructured road spectrum.In the in-situ road spectrum collection implementation,the route length limitation and road surface diversity may decrease the spectrum accuracy.The moving block bootstrap confidence interval estimation method is adopted to estimate the interval of the stable distribution parameters.According to both the stable distribution random series generation theory and the simulation capability of road simulation test bench,the reconstructed spectrum is composed by the amplitude truncated stable distribution random series.The validity and practicality of proposed methods are verified by reconstructing the spectrum of an off-highway vehicle riding through a test ground.An effect time domain road spectrum reconstruction method is provided for the unstructured road simulation and virtual reliability test.

Dynamic GSCA (Generalized Structured Component Analysis) with Applications to the Analysis of Effective Connectivity in Functional Neuroimaging Data

We propose a new method of structural equation modeling (SEM) for longitudinal and time series data, named Dynamic GSCA (Generalized Structured Component Analysis). The proposed method extends the original GSCA by incorporating a multivariate autoregressive model to account for the dynamic nature of data taken over time. Dynamic GSCA also incorporates direct and modulating effects of input variables on specific latent variables and on connections between latent variables, respectively. An alternating least square (ALS) algorithm is developed for parameter estimation. An improved bootstrap method called a modified moving block bootstrap method is used to assess reliability of parameter estimates, which deals with time dependence between consecutive observations effectively. We analyze synthetic and real data to illustrate the feasibility of the proposed method.

FixedbSubsampling and the Block Bootstrap: Improved Confidence Sets based onp-Value Calibration

SummarySubsampling and block-based bootstrap methods have been used in a wide range of inference problems for time series. To accommodate the dependence, these resampling methods involve a bandwidth parameter, such as the subsampling window width and block size in the block-based bootstrap. In empirical work, using different bandwidth parameters could lead to different inference results, but traditional first-order asymptotic theory does not capture the choice of the bandwidth. We propose to adopt the fixed b approach, as advocated by Kiefer and Vogelsang in the heteroscedasticity–auto-correlation robust testing context, to account for the influence of the bandwidth on inference. Under the fixed b asymptotic framework, we derive the asymptotic null distribution of the p-values for subsampling and the moving block bootstrap, and further propose a calibration of the traditional small-b-based confidence intervals (regions or bands) and tests. Our treatment is fairly general as it includes both finite dimensional parameters and infinite dimensional parameters, such as the marginal distribution function. Simulation results show that the fixed b approach is more accurate than the traditional small b approach in terms of approximating the finite sample distribution, and that the calibrated confidence sets tend to have smaller coverage errors than the uncalibrated counterparts.

Necessary and sufficient conditions for the moving blocks bootstrap central limit theorem of the mean

For strictly stationary sequences X i , we study the relations between statistics Z n =n −1/2∑ (X i −EX i ) and its Bootstrap counterpart We establish sufficient and necessary conditions for Central Limit Theorem in this setting and generalise the cases for which converges to Gaussian limit, while Z n fails to do so. In addition, we establish Bootstrap Central Limit Theorem without mixing conditions and identify the cases for which and Z n are asymptotically close, while both fail to converge in distribution.

Using the autoregressive conditional duration model to analyse the process of default contagion

Credit events are not independent, and the contagion effect is very common. The seriousness of the contagion effect depends on the change in the default contagion duration before and after credit events. This study uses the Autoregressive Conditional Duration (ACD) model to capture the durations of a series of credit events and to study the characteristics of a default duration series. The empirical samples are listed and Over-The-Counter (OTC) companies in Taiwan. The Moving Block Bootstrap (MBB) in Liu and Singh (1992) is employed to copy the sample data. The sample period is from October 1982 to December 2007. The results show that, in the entire sample and subsamples of the electronic information industry and construction industry, the default duration series demonstrates the conditional autocorrelation and cluster effect. The ACD model helps capture the contagion effect of credit events.

Assessing temporal representativeness of water quality monitoring data

The effectiveness of different monitoring methods in detecting temporal changes in water quality depends on the achievable sampling intervals, and how these relate to the extent of temporal variation. However, water quality sampling frequencies are rarely adjusted to the actual variation of the monitoring area. Manual sampling, for example, is often limited by the level of funding and not by the optimal timing to take samples. Restrictions in monitoring methods therefore often determine their ability to estimate the true mean and variance values for a certain time period or season. Consequently, we estimated how different sampling intervals determine the mean and standard deviation in a specific monitoring area by using high frequency data from in situ automated monitoring stations. Raw fluorescence measurements of chlorophyll a for three automated monitoring stations were calibrated by using phycocyanin fluorescence measurements and chlorophyll a analyzed from manual water samples in a laboratory. A moving block bootstrap simulation was then used to estimate the standard errors of the mean and standard deviations for different sample sizes. Our results showed that in a temperate, meso-eutrophic lake, relatively high errors in seasonal statistics can be expected from monthly sampling. Moreover, weekly sampling yielded relatively small accuracy benefits compared to a fortnightly sampling. The presented method for temporal representation analysis can be used as a tool in sampling design by adjusting the sampling interval to suit the actual temporal variation in the monitoring area, in addition to being used for estimating the usefulness of previously collected data.